Finite Group Databasebeta

← #34099324

Subgroups — #34099324

Order 20096.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758008C211yesmaximal, characteristic
2#23757620C211yesmaximal, characteristic
3#23757559C211yesmaximal, characteristic, complemented
4#23757483C211yesmaximal, characteristic, complemented
5#23757016C211yesmaximal, characteristic
6#23757005C211yesmaximal, characteristic, complemented
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578944—21no—
9#19097771C2211yescharacteristic
10#19097768C2211yescharacteristic
11#19097760—21no—
12#19097759—21no—
13#19097756—22no—
14C4×Dic314C2211yescharacteristic
15C4×D628—21no—
16#19097733—21no—
17#19097703—21no—
18C4⋊#11929434C2211yescharacteristic
19#19097696—21no—
20#19097694—21no—
21#19097693—22no—
22C4×Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×D628—21no—
28C2×D628—21no—
29C2×D628D412yes—
30C2×C4×D157—22no—
31#14578944—42no—
32C2×#11929434—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157D411yescharacteristic
36#14578933—21no—
37#14578931—22no—
38#14578931—22no—
39#14578931—22no—
40C4×Dic157—21no—
41C2×Q8×C157D411yescharacteristic
42C2×C8×C157—21no—
43C2×C8×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157D411yescharacteristic
46C42×C157—21no—
47C42×C157C2311yescharacteristic
48C22×D157—22no—
49Dic314—21no—
50Dic314—21no—
51Dic314—41no—
52Dic314—41no—
53D628—42no—
54D628—41no—
55D628—41no—
56D628—41no—
57D628—41no—
58C2×Dic157—21no—
59C2×Dic157—21no—
60C2×Dic157—41no—
61C4×D157—82no—
62#11929434—41no—
63Q8×C157—21no—
64Q8×C157—21no—
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157C2×D411yescharacteristic
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C8×C157—41no—
71C8×C157—41no—
72C22×C157#12011yescharacteristic
73C2×D157—42no—
74C2×D157—42no—
75C2×D157—42no—
76C4×C157—21no—
77C4×C157—41no—
78C4×C157—41no—
79C4×C157#13511yescharacteristic
80C4×C157#13511yescharacteristic
81C4×C157AGL(1,ℤ/8)11yescharacteristic
82C4×C157#13711yescharacteristic
83Dic157—41no—
84Dic157—41no—
85Dic157—81no—
86C2×C157#44911yescharacteristic
87C2×C157#45211yescharacteristic
88C2×C157#45511yescharacteristic
89D157—82no—
90C157#164011yescharacteristic, complemented, Sylow
91#1640—1571nomaximal, Sylow
92#559—1571no—
93#486—1571no—
94#475—1571no—
95#462—1571no—
96#331—1571no—
97Q8⋊C8—1571no—
98Q8⋊C8D15711yescharacteristic, complemented
99C2×Q16—3141no—
100C4⋊Q8—1571no—
101#127—1571no—
102#124—3141no—
103#124—3141no—
104#123—3142no—
105C4×Q8—1571no—
106C4×Q8C2×D15711yescharacteristic
107C4×D4—3141no—
108C4×D4—3141no—
109#106—3141no—
110C4⋊C8—1571no—
111C4⋊C8C2×D15711yescharacteristic
112#103—3141no—
113#103—3141no—
114#102—3142no—
115C4×C8C2×D15711yescharacteristic
116C2×Q8—3141no—
117C2×Q8—1571no—
118C2×Q8D62811yescharacteristic
119C2×D4—3141no—
120C2×D4—3141no—
121C2×D4—1572no—
122C22×C4—3142no—
123Q16—6282no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—3141no—
127C4⋊C4—21no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4—3141no—
131C4⋊C4—1571no—
132C4⋊C4D62811yescharacteristic
133C22⋊C4—3142no—
134C22⋊C4—3142no—
135C22⋊C4—3142no—
136C42—21no—
137C42—3141no—
138C42C22×D15711yescharacteristic
139C23—3142no—
140Q8—21no—
141Q8—21no—
142Q8—3141no—
143Q8—3141no—
144Q8—6281no—
145Q8—6281no—
146D4—6282no—
147D4—6281no—
148D4—6281no—
149D4—6281no—
150D4—6281no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3141no—
154C2×C4—3141no—
155C2×C4—6281no—
156C2×C4—12562no—
157C2×C4D4×D15711yescharacteristic
158C2×C4D4×D15711yescharacteristic
159C2×C4C2×D62811yescharacteristic
160C8—41no—
161C8—41no—
162C8—6281no—
163C22—6282no—
164C22—6282no—
165C22—6282no—
166C22#1909774111yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—6281no—
171C4—6281no—
172C4—12561no—
173C4#1909779211yescharacteristic
174C4#1909779311yescharacteristic
175C4#1909779911yescharacteristic
176C4#1909780311yescharacteristic
177C2—12562no—
178C2#2375742011yescharacteristic
179C2#2375743811yescharacteristic
180C2#2375745211yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingQ8⋊C8×C157C2
socleC22×C157#120
prime radical#19097768C22
hypercenterQ8⋊C8D157
hypocenterC157#1640
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #23757559C2semidirect
4. #23757483C2semidirect
6. #23757005C2semidirect
7. Q8⋊C8×C157C2semidirect
90. C157#1640semidirect
98. Q8⋊C8D157semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237580082
#237576202
#237575592
#237574832
#237570162
#237570052
Q8⋊C8×C1572
#1640157